Question

Let $ x + y + 1 = 0 $ and $ x - y + 4 = 0 $ be the asymptotes of a hyperbola $ H $. If $ (1, 1) $ is a point on $ H $, then the length of the latus rectum of $ H $ is

• A. \( 4\sqrt{3} \)
• B. \( \sqrt{3} \)
• C. \( \sqrt{2} \)
• D. \( \sqrt{5} \)

Hint:

For hyperbolas, use the equations of the asymptotes to determine the relationship between the constants \( a \) and \( b \), and then calculate the length of the latus rectum using the formula \( \frac{2b^2}{a} \).

Answer 1

The Correct Option is A

The given equation of the hyperbola is formed by the asymptotes: \[ x + y + 1 = 0 \quad \text{and} \quad x - y + 4 = 0 \] These represent the equations of the asymptotes of the hyperbola. 
Step 1: The general equation of the hyperbola is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] The asymptotes of the hyperbola are given by the equations: \[ y = \pm \frac{b}{a} x \] From the given equations of the asymptotes, we can determine the relationship between \( a \) and \( b \). 
Step 2: Since the length of the latus rectum of a hyperbola is given by \( \frac{2b^2}{a} \), we calculate this using the known values of \( a \) and \( b \) based on the asymptotes. 
The final result for the length of the latus rectum is \( 4\sqrt{3} \).